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Dr. Katz DEq Homework Solutions 29

# Dr. Katz DEq Homework Solutions 29 - spectrum Of a field K...

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Algebraic Solutions of Differential Equations 29 (2.3.4.2.5) Furthermore, h (i + 1) is an isomorphism if and only if H-W (i + 1) is an isomorphism. Proof Follows formally from the diagram. (2.3.4.3) Corollary. Assumptions as in (2.3.2), the canonical map (2.3.4.3.1) F~o~ i • F' +1 ~ Rnf~ (~-~t'lS (log D)) is an isomorphism if and only if the map h(i) (2.3.4.2.1) (2.3.4.3.2) h(i): F~o~i---~ R"f,(f2~/s(logD))/F '+1 is an isomorphism. If this is the case, then the canonical map (2.3.4.3.3) F,o~ ,- 1 F' + 2 ~ Rnf, (~"~t'/s (log D)) is an isomorphism if and only if the i+ l'st Hasse-Witt operation (which is defined, because h(i) is supposed an isomorphism) (2.3.4.3.4) H-W (i + 1): F~s R"-'- 'f, (f21~s ~ (log D)) ~ R"-'- - if, (f2~-s ~ (log D)) is an isomorphism. Proof. The second equivalence follows from the first (applied to i + t), in virtue of (2.3.4.2.5). Precisely as in the proof of (2.3.4.1.7), the proof of the first equivalence is immediately reduced to the case in which S is the
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Unformatted text preview: spectrum Of a field K. Thanks to (2.3.4.1.11), the homomorphism (2.3.4.3.1) has source and target of the same dimension, as does (2.3.4.3.2), and both homomorphisms have the same kernel, namely F i+1(-3 fcnon i. Q.E.D. (2.3.4.4) Corollary. Assumptions as in (2.3.2), and n>O fixed, suppose that the map h(i) figuring in (2.3.4.2.4) is an isomorphism. Then: (2.3.4.4.1) The Cs-module u+a~F~ox~-t(R"f,(f2}/s(logD))) is locally free, and its formation commutes with arbitrary change of base S'--~ S. (2.3.4.4.2) The canonical mapping F~on i- 1 RnJ, (f2~c/s (log D)) ,- FZo ~ ' @ F i + 1 ~ FZo- ~ i- 1 is an isomorphism. Proof. It suffices to prove (2.3.4.4.2), since by hypothesis F~o ~ ~-~/F~o: i is locally free, and its formation commutes with arbitrary change of base S'-~ S. The composite mapping, formed from the bottom half of the diagram (2.3.4.2.4), F:o~ '-1 (R" f, (t2~:/s (log D))) hi, +x), R"f, (~]:s (log D))/F '+ 2 F~o-~ '(R"J, (Q'x/s (log D))) ~hc0 -~ R.f, (f2~,/s (log D))/F '+1...
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• Fall '11
• NormanKatz
• Equivalence relation, Isomorphism, Homomorphism, Congruence relation, Algebraic Solutions of Differential Equations

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